I've never quite understood the theory behind why dark matter and dark energy exist. I know it has something to do with gravitational pull being stronger than what we calculate it SHOULD be, could someone explain this in more detail?

3 Answers
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In fact, we do not know that dark matter and dark energy do exist, but we have indirect clues. You will often see claims that dark matter and dark energy are two of the major problems of cosmology today, including by professional astronomers, but this is an epistemological misconception: you cannot call a hypothesis a problem.

Currently we have several observational anomalies (as defined per T. Kuhn): namely the rotation curve of galaxies, the luminosity of supernovae, the power spectrum of the CMB... (and some additional things like the difficulty to measure G or to detect directly gravitational waves).

According to the history of science, we have 3 ways to solve these anomalies:

ontological approaches (dark matter and dark energy)

legislatives approaches (extensions of General Relativity)

paradigmatic approaches (inhomogeneous Universes)

Dark matter and dark energy are two ontological approaches that seems to work pretty well: by adding two simple components we can fit a wide range of observations. This is why astronomers like them: their modelization are relatively easy and allow to fit everything well. But for the moment we do not know their nature.

Some say that with WMAP (Wilkinson Microwave Anisotropy Probe) and Planck space observatory we have entered in the era of precision cosmology: but this is only true if our paradigm of dark matter/dark energy approaches is correct. If this is not the case, we currently do not interpret correctly what we measure, but ultimately more accurate measures can lead to a better view of current cosmological anomalies.

So dark matter and dark energy are two elegant and minimalist approaches to several fundamental problems. This is a very hot topic in cosmology, but we may realize, one day, that we only had problems because our current paradigm was erroneous.

Precision cosmology simply means we are measuring phenomena to higher and higher precision. In the case of the microwave background, it's 1 part in about 100,000. The accuracy of measuring the components of the universe, among many other cosmological parameters, are not as good as that (regardless of what you assume about dark matter and dark energy).
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astromaxDec 6 '13 at 21:27

Dark matter and dark energy are two different things, accounting for different observations.

Dark matter:

Dark matter is needed to explain, among other things, the rotation curve of galaxies. One could expect these rotation curves to decrease at large radii (because one should expect keplerian rotation for galaxies), and it is not the case, the rotation curves are flat, that accounts for a "hidden mass" in galaxies to sustain such a rotation pattern, that is the dark matter. Dark matter also accounts for some galaxy clusters properties, and CMB fluctuations.

Dark energy:

Dark energy is needed to explain the accelerated expansion of the Universe. This expansion was mostly measured using Type Ia supernovæ (thermonuclear supernovæ, that are the results of a white dwarf accreting matter and reaching Chandrasekhar mass) that are "standard candles". The absolute magnitude of these objects are thought to be well-known, so if you observe them in the Universe, you can easily deduce their distance from their redshift. These observations show an accelerated expansion of the Universe (by expansion, we mean that the relative distance between two distant galaxies is increasing, distance in the local Universe are not affected). To account for this acceleration, one needs some energy uniformly distributed in the Universe, that is the dark energy.

Don't be confused:

Dark energy and dark matter cannot be mistaken one for the other, because dark energy is uniformly distributed in the Universe, whereas dark matter follows some distribution (it is generally more or less distributed with the matter).

Final words:

Finally, "dark matter" and "dark energy" are mostly words to characterize our ignorance on these observations that cannot be explained by standard laws of physics. That is why there is such intense work on these subjects, to try to characterize these dark things better and to understand what they could be.

People have a pretty good idea that dark energy exist because, when you chart the expansion of two objects in the universe over time, from its origin, there is a bell curve. Basically, the speed of the universe's expansion started out faster, than slowed down, and recently (or at least relatively), the expansion has sped back up again. This implies that when the Universe began, it was rapidly expanding, and as this momentum began to flag, gravity began to pull things back together. However, to explain the recent acceleration of expansion, there has to be a force that has increased in strength over time and has been able to cause acceleration in expansion (all of this occurs between galaxies, in areas like solar systems and galaxies, gravity is much to strong to be affected). It is believed that as the universe expands, dark energy increases uniformly to occupy the new space. This means that dark energy increases, thus accelerating expansion.

We know that the universe is expanding because of red shift. Basically, as light travels through the universe, the universe is expanding. Over enough distance, the light will perceptibly lose energy, so that violet light will fall towards red (hence the name red shift). This is because, viewed from our expanding universe, the light appears to be losing energy, when really it is having to travel further as space expands.

As for dark matter, in short, we know that there is something that has gravity that we cannot detect otherwise. Dark matter, unlike dark energy, is not uniform. It forms strand like formations, which correspond to the strand like formations of galaxies.